cotangent bundle


Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



Given a differentiable manifold XX, the cotangent bundle T *(X)T^*(X) of XX is the dual vector bundle over XX dual to the tangent bundle TxT x of XX.

A cotangent vector or covector on XX is an element of T *(X)T^*(X). The cotangent space of XX at a point aa is the fiber T a *(X)T^*_a(X) of T *(X)T^*(X) over aa; it is a vector space. A covector field on XX is a section of T *(X)T^*(X). (More generally, a differential form on XX is a section of the exterior algebra of T *(X)T^*(X); a covector field is a differential 1-form.)

Given a covector ω\omega at aa and a tangent vector vv at aa, the pairing ω,v\langle{\omega,v}\rangle is a scalar (a real number, usually). This (with some details about linearity and universality) is basically what it means for T *(X)T^*(X) to be the dual vector bundle to T *(X)T_*(X). More globally, given a covector field ω\omega and a tangent vector field vv, the paring ω,v\langle{\omega,v}\rangle is a scalar function on XX.

Given a point aa in XX and a differentiable (real-valued) partial function ff defined near aa, the differential d af\mathrm{d}_a f of ff at aa is a covector on XX at aa; given a tangent vector vv at aa, the pairing is given by

d af,v=v[f], \langle{\mathrm{d}_a f, v}\rangle = v[f] ,

thinking of vv as a derivation on differentiable functions defined near aa. (It is really the germ at aa of ff that matters here.) More globally, given a differentiable function ff, the de Rham differential df\mathrm{d}f of ff is a covector field on XX; given a vector field vv, the pairing is given by

df,v=v[f], \langle{\mathrm{d}f, v}\rangle = v[f] ,

thinking of vv as a derivation on differentiable functions.

One can also define covectors at aa to be germs of differentiable functions at aa, modulo the equivalence relation that d af=d ag\mathrm{d}_a f = \mathrm{d}_a g if fgf - g is constant on some neighbourhood of aa. In general, a covector field won't be of the form df\mathrm{d}f, but it will be a sum of terms of the form hdfh \mathrm{d}f. More specifically, a covector field ω\omega on a coordinate patch can be written

ω= iω idx i \omega = \sum_i \omega_i\, \mathrm{d}x^i

in local coordinates (x 1,,x n)(x^1,\ldots,x^n). This fact can also be used as the basis of a definition of the cotangent bundle.


Symplectic structure

Every cotangent bundle T *XT^\ast X carries itself a canonical differential 1-form

θΩ 1(T *X) \theta \in \Omega^1(T^* X)

with the property that under the isomorphism

j:Γ(T *X)Ω 1(X) j \;\colon\; \Gamma(T^* X) \stackrel{\simeq}{\to} \Omega^1(X)

between differential 1-forms and smooth sections of the cotangent bundle we have for every smooth section σΓ(T *X)\sigma \in \Gamma(T^* X) the identification

σ *θ=j(σ) \sigma^* \theta = j(\sigma)

between the pullback of θ\theta along σ\sigma and the 1-form corresponding to σ\sigma under jj.

This unique differential 1-form θΩ 1(T *X)\theta \in \Omega^1(T^* X) is called the Liouville-Poincaré 1-form or canonical form or tautological form on the cotangent bundle.

The de Rham differential ωdθ\omega \coloneqq d \theta is a symplectic form. Hence every cotangent bundle is canonically a symplectic manifold.

On a coordinate chart n\mathbb{R}^n of XX with canonical coordinate functions denoted (x i)(x^i), the cotangent bundle over the chart is T * n n× nT^\ast \mathbb{R}^n \simeq \mathbb{R}^n \times \mathbb{R}^n with canonical coordinates ((x i),(p j))((x^i), (p_j)). In these coordinates the canonical 1-form is (using Einstein summation convention)

θ=p idx i \theta = p_i d x^i

and hence the symplectic form is

ω=dp idq i. \omega = d p_i \wedge d q^i \,.

Revised on November 23, 2017 07:48:04 by Urs Schreiber (